Convergence of One-Parameter Operator Semigroups: In Models of Mathematical Biology and Elsewhere

454 Pages·2016·13.154 MB·English

by  Adam Bobrowski

ConvergenceofOne-ParameterOperatorSemigroups
This book presents a detailed and contemporary account of the classical the-
ory of convergence of semigroups and its more recent developments treating
the case where the limit semigroup, in contrast to the approximating semi-
groups,actsmerelyonasubspaceoftheoriginalBanachspace(thisisthecase,
for example, with singular perturbations). The author demonstrates the far-
reachingapplicationsofthistheoryusingrealexamplesfromvariousbranches
of pure and applied mathematics, with a particular emphasis on mathemati-
calbiology.Theseexamplesalsoserveasshort,nontechnicalintroductionsto
biologicalconceptsinvolved,allowingreaderstodevelopintuitionsunderlying
mathematicalresults.
Thebookmayserveasausefulreference,containingasignificantnumber
ofnewresultsrangingfromtheanalysisoffishpopulationstosignalingpath-
waysinlivingcells.Itcomprisesmanyshortchapters,whichallowsreadersto
pickandchoosethosetopicsmostrelevanttothem,anditcontains160end-of-
chapterexercisessothatreaderscantesttheirunderstandingofthematerialas
theygoalong.
Adam Bobrowski isaprofessorandChairmanoftheDepartmentofMath-
ematics at Lublin University of Technology, Poland. He has authored over
50 scientific papers and two books Functional Analysis for Probability and
StochasticProcessesandAnOperatorSemigroupinMathematicalGenetics.
NEW MATHEMATICAL MONOGRAPHS
EditorialBoard
BélaBollobás,WilliamFulton,AnatoleKatok,FrancesKirwan,PeterSarnak,
BarrySimon,BurtTotaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity
Press.Foracompleteserieslisting,visitwww.cambridge.org/mathematics.
1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2. J.B.GarnettandD.E.MarshallHarmonicMeasure
3. P.CohnFreeIdealRingsandLocalizationinGeneralRings
4. E.BombieriandW.GublerHeightsinDiophantineGeometry
5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7. A.ShlapentokhHilbert’sTenthProblem
8. G.MichlerTheoryofFiniteSimpleGroupsI
9. A.BakerandG.WüstholzLogarithmicFormsandDiophantineGeometry
10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13. M.GrandisDirectedAlgebraicTopology
14. G.MichlerTheoryofFiniteSimpleGroupsII
15. R.SchertzComplexMultiplication
16. S.BlochLecturesonAlgebraicCycles(2ndEdition)
17. B.Conrad,O.GabberandG.PrasadPseudo-ReductiveGroups
18. T.DownarowiczEntropyinDynamicalSystems
19. C.SimpsonHomotopyTheoryofHigherCategories
20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI
21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII
22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory
23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry
24. E.RiehlCategoricalHomotopyTheory
25. B.A.MunsonandI.Volic´CubicalHomotopyTheory
26. B.Conrad,O.GabberandG.PrasadPseudo-ReductiveGroups(2ndEdition)
27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetric
MeasureSpaces
28. Y.-G.OhSymplecticTopologyandFloerHomologyI
29. Y.-G.OhSymplecticTopologyandFloerHomologyII
30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups
Convergence of One-Parameter Operator
Semigroups
In Models of Mathematical Biology and Elsewhere
ADAM BOBROWSKI
LublinUniversityofTechnology,Poland
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107137431
©AdamBobrowski2016
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2016
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-107-13743-1Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
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Tothemostenthusiasticdrummer,boxer,andeaterever–myson
Marek

Contents
Preface pagexi
1 SemigroupsofOperatorsandCosineOperatorFunctions 1
PART I REGULAR CONVERGENCE
2 TheFirstConvergenceTheorem 13
3 ContinuousDependenceonBoundaryConditions 17
4 SemipermeableMembrane 24
5 ConvergenceofForms 32
6 UniformApproximationofSemigroups 40
7 ConvergenceofResolvents 45
8 (Regular)ConvergenceofSemigroups 51
9 AQueueinHeavyTraffic 56
10 ElasticBrownianMotions 60
11 BacktotheMembrane 65
12 TelegraphwithSmallParameter 70
13 MinimalMarkovChains 74
14 OutsideoftheRegularitySpace:ABird’s-EyeView 81
15 Hasegawa’sCondition 85
16 Blackwell’sExample 90
viii Contents
17 Wright’sDiffusion 96
18 Discrete-TimeApproximation 100
19 Discrete-TimeApproximation:Examples 105
20 BacktoWright’sDiffusion 112
21 Kingman’sn-Coalescent 116
22 TheFeynman–KacFormula 121
23 TheTwo-DimensionalDiracEquation 128
24 ApproximatingSpaces 132
25 Boundedness,Stabilization 136
PART II IRREGULAR CONVERGENCE
26 FirstExamples 145
27 ExtremelyStrongGeneticDrift 152
28 TheNatureofIrregularConvergence 157
29 IrregularConvergenceIsPreservedUnder
BoundedPerturbations 163
30 Stein’sModel 166
31 UniformlyHolomorphicSemigroups 171
32 AsymptoticBehaviorofSemigroups 177
33 FastNeurotransmitters 189
34 FastNeurotransmittersII 197
35 FromDiffusionsonGraphstoMarkovChainsand
BackAgain 203
36 SemilinearEquations,EarlyCancerModeling 210
37 Coagulation-FragmentationEquation 219
38 HomogenizationTheorem 228
39 ShadowSystems 236
Contents ix
40 Kinases 241
41 UniformlyDifferentiableSemigroups 250
42 Kurtz’sSingularPerturbationTheorem 253
43 ASingularlyPerturbedMarkovChain 258
44 ATikhonov-TypeTheorem 263
45 FastMotionandFrequentJumpsTheoremsforPiecewise
DeterministicProcesses 271
46 ModelsofGeneRegulationandGeneExpression 281
47 Oligopolies,ManufacturingSystems,andClimateChanges 287
48 ConvexCombinationsofFellerGenerators 292
49 TheDorrohTheoremandtheVolkonskiiFormula 299
50 ConvexCombinationsinBiologicalModels 303
51 Recombination 311
52 Recombination(Continued) 318
53 AveragingPrincipleofFreidlinandWentzell:
Khasminskii’sExample 327
54 ComparingSemigroups 335
55 RelationstoAsymptoticAnalysis 341
56 Greiner’sTheorem 345
57 FishPopulationDynamicsandConvexCombinationof
BoundaryConditions 352
58 AveragingPrincipleofFreidlinandWentzell:Emergenceof
TransmissionConditions 361
59 AveragingPrincipleContinued:L1-Setting 370
PART III CONVERGENCE OF COSINE FAMILIES
60 RegularConvergenceofCosineFamilies 383
61 CosinesConvergeinaRegularWay 390